Asymptotic fingerprinting capacity in the Combined Digit Model

1. Asymptotic fingerprinting capacity in the Combined Digit Model Dion Boesten and Boris Škorić

2. Outline • forensic watermarking • collusion attack models: Restricted Digit Model and Combined Digit Model • bias-based codes • fingerprinting capacity • large coalition asymptotics • Previous results: Restricted Digit Model • New contribution: Combined Digit Model

3. Forensic watermarking originalcontent originalcontent watermarked content unique watermark unique watermark Detector Embedder Attack

4. Collusion attacks • Simplifying assumption: segments into which q-ary symbols can be embedded m content segments n users collusion attack: c attackers pool their resources

5. Attack models: Restricted Digit Model (RDM) • "Marking assumption": can't produce unseen symbol • Restricted Digit Model:choose from available symbols m content segments c attackers allowed symbols

6. Attack models: Combined Digit Model (CDM) • More realistic • Allows for signal processing attacks • mixing • noise [BŠ et al. 2009] symbol detectionprobability: t |ψ| 1-t detected:W alphabetQ |ψ| mixed:Ψ⊆Ω receivedΩ⊆Q 1-r attack r Noise parameter r. Mixing parameters t1 ≥ t2 ≥ t3 ...

7. Bias-based codes [Tardos 2003] content segments • Code generation • Biases drawn from distribution F • Code entries generated per segment j using the bias: Pr[Xij = α] = pjα. • Attack • Coalition size c. • Same strategy in each segment • In Combined Digit Model:strategy = choice of subset Ψ⊆Ω,possibly nondeterministic. • Accusation • algorithm for finding at least one attacker,based on distributed and observed symbols. symbol biases Ω={A,B}Allowed Ψ: {A}, {B}, {A,B}

8. Collusion attack viewed as malicious noise • Noisy communication channel • From symbol embedding to detection • Coalition attack causes "noise" • Channel capacity • Apply information theory • Rate of a tracing code:R = (logq n)/m • Capacity C = max. achievable rate. Fundamental upper bound. • Results for Restricted Digit Model, and #attackers → ∞ • Huang&Moulin 2010Binary codes (q=2): • Boesten&Škorić 2011Arbitrary alphabet size: n = #usersm = #segments q = alphabet size

9. Capacity for the Combined Digit Model • The math • Look at one segment • Define counters Σα= #attackers who receive α • Parametrization of the attack strategy: • Capacity: p = bias vector F = prob. density for p W = set of detected symbols I(W;Σ) H(Σ) H(W) - + θ F

10. CDM capacity: further steps • Apply Sion's theorem • "Value" of max-min and min-max game is the same! • Limit c → ∞: • Σ very close to cp • Taylor expansion in Σ/c– p • Re-paramerization • γ: mapping from q-dim. hypersphereto (2q-1)-dim. hypersphere. • Jacobian J • Pay-off function Tr(JTJ)

11. CDM capacity: constraints • Looks like beautiful math, but ... nasty constraint on the mapping γ • We did not dare to try q>2 • Binary case: Constrained geodesics

12. CDM capacity: numerical results for q=2 • Part of the graphs we understand intuitively • Stronger attack options => lower capacity • Near (r=0, t1=1) RDM-like behaviour; weak dependence on t2 • Away from RDM we have little intuition

13. Summary • Asymptotic capacity for the Combined Digit Model • Partly the same exercise as in Restricted Digit Model • Find optimal hypersphere → hypersphere mapping • But ... • higer-dimensional space • nasty constraint on the mapping • Numerics for binary alphabet • constrained geodesics in 2 dimensions • graphs show how attack parameters (r, t1, t2) affect capacity • useful for code design • Future work (perhaps ...) • change attack model to get analytic results

14. Questions?